Heredity73 (1994) 657—679 Received9 March 1994 Genetical Society of Great Britain Review article Developments in the prediction of effective population size ARMANDO CABALLERO Institute of Cell, Animal and Population Biology, University of Edinburgh, West Mains Road, Edinburgh EH9 3JT, UK Effectivepopulation size is a key parameter in evolutionary and quantitative genetics because it measures the rate of genetic drift and inbreeding. Predictive equations of effective size under a range of circumstances and some of their implications are reviewed in this paper. Derivations are made for the simplest cases, and the inter-relations between different formulae and methods are discussed. Keywords: heterozygosity, inbreeding, nonrandom mating, population numbers, random genetic drift. Contents 1 The idealized population and effective population size 2 Self-fertilization not allowed 3 Different numbers of male and female parents 4 Variable population size over generations 5 Nonrandom contribution from parents 5.1 Variation due to noninhented causes 5.1.1 Monoecious diploid species with selfing allowed 5.1.2 Separate sexes 5.1.3 Haploid and polyploid species 5.1.4 X-linked genes 5.1.5 Systems of mating 5.2 Variation due to inherited causes 5.2.1 Effective size from the variance of change in gene frequency over generations 5.2.2 Selection on fitness 5.2.3 Effective size from long-term contributions of ancestors 5.2.4 Index selection 6 Nonrandom mating with population subdivision 7 Overlapping generations 8 Prospects for future developments References 1 Theidealized populationand effective thefinite sampling of gametes. These erratic changes populationsize constitutethe so-called dispersive process or genetic In an infinitely large population and in the absence of drift, which is likely to be one of the main factors mutation, migration and selection, gene and genotype governing molecular evolution (Kimura, 1983) and has frequencies remain constant over generations. In finite implications on the rate and time to fixation of select- populations, however, gene frequencies fluctuate ively advantageous or deleterious genes (see Crow & randomly from generation to generation as a result of Kimura, 1970, chap. 8). 657 658 A. CABALLERO Geneticdrift, at least in unstructured populations, is where quantified by a single parameter, effective population size, which can be computed or estimated from labora- AF=Fh1 tory or field data, and predicted under a range of 1—F,1 circumstances. This paper reviews predictive equations of the effective size and their inter-relations. For The observable consequence of this increase in completeness and for a better understanding of the inbreeding is a reduction in the expected hetero- concepts, some of the basic theory is summarized first. zygosity (H) each generation, The simplest possible conditions under which the dispersive process can be studied are met in the Wright—Fisher idealized population (Fisher, 1930; (3) Wright, 1931). This consists of an infinite, randomly mated base population subdivided into infinitely many or, relative to that in the base population, subpopulations, each with a constant number, N, of breeding individuals per generation, In each subpopu- lation, parents produce an infinite number of male and (4) female gametes into a large pool from which only 2N are sampled and united to produce the N zygotes of the =1—1/2Nis the largest nonunit eigenvalue of the following generation. All individuals survive from birth transition matrix to adulthood. Both the sampling of gametes and their union (including self-fertilization) are random, so that 2N i 2N-j all parents have an equal chance of producing offspring I and the distribution of offspring number is multi- nomial. Systematic changes in allele frequencies are which,for the various possible states (number of excluded in this idealized population, generations do copies, i, of a given allele) in generation t —1,gives the not overlap, and only autosomal loci are considered. probability of each state (j) in generation t. The dispersive process can be studied as a sampling The relationship between the variance of gene process or as an inbreeding process because both an frequency over subpopulations and the coefficient of increase in the variance of gene frequency among sub- inbreeding is: populations and an increase in homozygosity occur as a result of the finite population size. Under the simple conditions of the idealized popula- u1=q(1—q) [i tion, sampling of gametes is binomial and the variance (_) ]=q(1_q)Ft, of the change in gene frequency is: where the one generation delay between a and F is due to the fact that drift begins one generation earlier 2 q(1—q) (1) than inbreeding. Thus, in N individuals randomly 2N chosen from an infinite population there is yet no inbreeding but there is drift. where q is the allele frequency of a gene in the infinite base population. The coefficient of inbreeding at It is obvious that real populations are very unlikely to meet the conditions of the idealized population generation t, the probability that two gametes which unite to produce a zygote in generation I carry identical defined above and, therefore, the number of breeding individuals does not describe appropriately the effects by descent copies of a gene (Wright, 1922; Malécot, 1948), is of inbreeding and gene frequency drift in most practi- cal situations. The concept of effective population size (Ne) was introduced by Sewall Wright (1931, 1938, F1=+(1_) F1, 1939) to overcome this problem and has been developed subsequently by others, mainly James F. where the first term denotes identity by descent from Crow and coworkers (Crow & Kimura, 1970, pp. copies of a gene of an individual in generation t —1and 345—364; Crow & Denniston, 1988). The effective size the second, that from copies of a gene of an individual of a population is defined as the size of an idealized in previous generations. The rate of increase in population which would give rise to the variance of inbreeding per generation is thus: change in gene frequency or the rate of inbreeding observed in the actual population under consideration, i.e. Ne=q(lq)/2aq or N=1/2F (from eqns (1) 2N (2) and (2)), which correspond to the so-called variance EFFECTIVE POPULATION SIZE 659 and inbreeding effective sizes, respectively. Thus, the the demographical parameters necessary to estimate effective size gives a measure of the rate of genetic drift Ne in natural populations are not dealt with in this and inbreeding in the population. As neutral genetic paper. Some of these problems and a comparison variation depends directly on these parameters, effec- between different methods of estimation are given by tive size gives a prediction of the impact of manage- Begon etal. (1980) (see also Nunney & Elam (1994)). ment practices on genetic variation. Also, as the A recent account of the practical and theoretical effective size becomes smaller, weakly selected alleles considerations of the estimation of Ne from temporal become effectively neutral. It is, therefore, a primary changes in gene frequency of allozyme markers is given variable to biologists concerned with monitoring by Waples (1989). genetic variation in natural populations and, particu- larly, those concerned with the management of endang- 2Self-fertilization not allowed ered or zoo species (Lande & Barrowclough, 1987; Nunney & Campbell, 1993). Effective size is also Thesimplest deviation from the idealized population important in plant and animal breeding because its which can be considered is the exclusion of self-fertili- magnitude affects the response to artificial selection zation in monoecious species. Under this situation, the and its variance (see Hill (1985a, 1986) for reviews). probability that two gametes which unite to produce a If the variance of change in gene frequency or the zygote in generation t carry identical copies of a gene rate of increase in inbreeding are known, because the of an individual in generation t —2 is (1 + genotypes can be distinguished and hence the geno- and the probability that they carry identical copies of a typic frequencies estimated or because pedigrees are gene from different individuals is (1 —1/N)F1.Thus, available, the effective population size can be estimated the inbreeding in generation t is or computed directly from the expressions above. For example, if we can trace an observable quantity such as (5) the heterozygosity so that we know its rate of decay F,=2+(1_) F1. we can use eqns (2) and (3) to estimate the asymptotic Ne. This is what is called effective size 'for In this case, genetic drift begins two generations earlier random extinction' (Crow (1954) after a result from than inbreeding. By using eqns (3) and (4), expression Haldane (1939)) or 'eigenvalue' effective size, because (5) leads to N.?.2—A(N— 1)— 1/2=0, with solution it is the result arising when the largest nonunit eigen- A=(N—1+ITf)/2N'11/(2N+1) (Wright, value of the transition matrix of a Markov Chain is 1969, p. 195). Thus, from eqn (3): obtained (see Ewens, 1979, 1982). The actual Ne can 1 also be computed from the inbreeding coefficients (6) obtained from pedigrees. Estimations can be made even when there are individuals with uncertain parent- age in the pedigree, by including information external and, by the definition of N and using eqn (2), to the records (Pérez-Enciso etal., 1992). N + 0.5, which is a very small difference if N is When information on genotypic frequencies or pedi- large. Note that this value is the eigenvalue effective grees is not available, effective size can still be pre- size because it is computed from the largest nonunit dicted under certain circumstances (in which one or eigenvalue ()ofthe transition matrix corresponding to more assumptions of the idealized population are this model (see Ewens, 1979, p. 107).